Name
Partial Differential Equations 3150
Midterm Exam 2
Exam Date: Wednesday, 9 April 2014
Instructions: This exam is timed for 50 mnimites. Up to 60 minutes is possible. No
calculators, notes, tables or books. No answer check is expected. Details count 3/4, answers
count 1/4.
Fourier Series
Problem 1.
(a) [20%] State the Fourier Convergence Theorem for a function 1(x) defined
on—L<x<L.
/
Graded Details: (1) Hypoth, (2) Conclu, (3) Serinula, (4) Cot
formulas.
A-
/
(b) [20%] True or False. Each part earns 5/20 if answered correctly and 2/20 if
answered incorrectly.
7-
True or False. Assume f(x) is periodic of period 5 and continuously differ
entiable on —cc < x < cc. Let F(x) be the formal Fourier series of 1(x) on
lxi
L, L
=
5/2. Then f(11.2)
=
F(1.2).
• True or False. Gibb’s overshoot could fail to happen at a jump discontinuity,
but when it happens the overshoot is about 9%.
—r •
True or False. The function 1(x)
of any period.
sin(2x) + sin(irx) is odd but not periodic
• True or False. The even periodic extension of f(x)
2 equals jx 6j on the interval 5 x
7.
=
x on 0 < x < 1 of period
—
.,t.ç(
(c) [60%] Let 1(x) be the even extension to
xi
< ir of the function sin(2x) on
has a formal Fourier cosine series F(x) on 0 <x < it, which
0 <x <it. Then f
is an even 2ir-periodic extension of 1(x) to —cc <x <cc.
L/ Make a graph of 1(x) on
—it
x
it.
Make a graph of F(x) over three periods.
• VVrite the series formula for F(x) and the Fourier cosine coefficient formula.
• Find an integral formula, or the exact value, of the first nonzero term in the
Fourier cosine series expansion F(x).
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3150 Exam 2 S2014
Name.
Wave Equation: The Finite String
Problem 2.
A(a) [50%] Display the series formula, complete with derivation details, for the
separation of variables solution u(:r, t.) of the finite string problem
u(x,t)
=
=
/
I
u(1,t)
v(x,0)
ut(x,0)
=
=
d(x,),
0< x < 1,
0,
0,
f(x),
g(x),
t >0,
t>0,
t>0,
0< x <1,
0<x<1.
Expected in the formula for u(x, t) are constants times product solutions (the normal
modes).
Graded Details: (1) Separation of variables, (2) Product solution boundary value
problem, (3) Product solution formulas,(4) Superposition details, (5) Series formula.
.
(b) [20%] Display Fourier coefficient formulas for the solution of part (a).
A- (c) [30%] Evaluate the Fourier coefficient formulas when
1
g(x)
=
50 on 0
)
x
,
g(x)
=
=
0 on 0 <x
<
1 and
‘)
—
XT
f(x)
0 otherwise.
x
1
1
.
tYo’
-
ft) o
(
0
I i4 C
L
4
r
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()
L= I
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S2014
3150 Exam 2
Name.
Fourier Transform
Problem 3.
to
k (a) [10%] Define Haherman’s Fourier transform pair.
(b) [30%] Assume f(r)
Fourier transform F(w)
2 on —1
of f(x).
=
<
x
0
<
and
f(x)
=
0
otherwise.
Compute
the
Graded Details: (1) Transform formula, (2) Integration details, (3) Answer.
,/ç(c)
[60%] The
heat kernel g(x)
and
the
error function erf(x)
are defined
by the
equations
g(x)
Solve
the
(4)
=
ve
dkt,
erf(x)=
2
f..2
e
dz.
infinite rod heat conduction problem
u(,t)
=
‘u(x,t),
—oo<x<oo,
u(r,0)
=
f(x),
—cx <x <oo,
f(x)
=
Graded Details:
use,
I2
t>0,
150 0<:v<1,
0
otherwise
(1) Fourier Transform method,
Error function methods,
(5) Final
(2)
Convolution,
(3)
Heat kernel
answer, expressed in terms of
the
function.
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